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Simplifying 4n2 = 8n + 1 Reorder the terms: 4n2 = 1 + 8n Solving 4n2 = 1 + 8n Solving for variable 'n'. Reorder the terms: -1 + -8n + 4n2 = 1 + 8n + -1 + -8n Reorder the terms: -1 + -8n + 4n2 = 1 + -1 + 8n + -8n Combine like terms: 1 + -1 = 0 -1 + -8n + 4n2 = 0 + 8n + -8n -1 + -8n + 4n2 = 8n + -8n Combine like terms: 8n + -8n = 0 -1 + -8n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.25 + -2n + n2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + -2n + 0.25 + n2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + -2n + n2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + -2n + n2 = 0 + 0.25 -2n + n2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 -2n + n2 = 0.25 The n term is -2n. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2n + 1 + n2 = 0.25 + 1 Reorder the terms: 1 + -2n + n2 = 0.25 + 1 Combine like terms: 0.25 + 1 = 1.25 1 + -2n + n2 = 1.25 Factor a perfect square on the left side: (n + -1)(n + -1) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (n + -1) equal to 1.118033989 and -1.118033989.Subproblem 1
n + -1 = 1.118033989 Simplifying n + -1 = 1.118033989 Reorder the terms: -1 + n = 1.118033989 Solving -1 + n = 1.118033989 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = 1.118033989 + 1 Combine like terms: -1 + 1 = 0 0 + n = 1.118033989 + 1 n = 1.118033989 + 1 Combine like terms: 1.118033989 + 1 = 2.118033989 n = 2.118033989 Simplifying n = 2.118033989Subproblem 2
n + -1 = -1.118033989 Simplifying n + -1 = -1.118033989 Reorder the terms: -1 + n = -1.118033989 Solving -1 + n = -1.118033989 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = -1.118033989 + 1 Combine like terms: -1 + 1 = 0 0 + n = -1.118033989 + 1 n = -1.118033989 + 1 Combine like terms: -1.118033989 + 1 = -0.118033989 n = -0.118033989 Simplifying n = -0.118033989Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.118033989, -0.118033989}
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